Problem when scaling on X axis
You seem to have some sort of mirroring applied: all your vertices on the left half of the mesh are just mirror images of the vertices on the right. Therefore, the median point of your selection is not at the middle of the mesh, because it only takes into account "real" vertices, the ones on the right. When you "Scale", your vertices will all move away from that point.

Make sure that your 3D cursor is somewhere on the middle plane (it looks like it already is), then select "3D Cursor" as your pivoting point:



Finally, you should be able to { S, X, move your mouse } to scale.